Characterizing Multigraded Regularity and Virtual Resolutions on Products of Projective Spaces

Abstract

We explore the relationship between multigraded Castelnuovo--Mumford regularity, truncations, Betti numbers, and virtual resolutions on a product of projective spaces $X$. After proving a uniqueness theorem for certain virtual resolutions, we show that the multigraded regularity region of a module $M$ is determined by the minimal graded free resolutions of the truncations $M_{\ge \mathbf{d}}$ for $\mathbf{d}\in \mathrm{Pic}X$. Further, by relating the minimal graded free resolutions of $M$ and $M_{\ge \mathbf{d}}$ we provide a new bound on multigraded regularity of $M$ in terms of its Betti numbers. Using this characterization of regularity and this bound we also compute the multigraded Castelnuovo--Mumford regularity for a wide class of complete intersections in products of projective spaces.

To appear in Advances in Mathematics. Section 3.1 is removed and Sections 3.2-3.4 are moved to the appendix